introduction to time-series analysis with pi system data...yeast can easily form a membrane that...
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Introduction to Time-Series Analysis with PI System Data
Prof. Brian D. Davison
Prof. Martin Takáč
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Presentation Outline
• Preliminaries: Sensors, Data set, and Sampling
• Task #1: Prediction of ADF
• Task #2: Modeling Cooling
• Task #3: PCA for Visualization
• Lessons Learned
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3
Preliminaries Sensors, Data set, and Sampling
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Sensor data
4
Common problems:
• missing values (wrong reading from sensor/communication issue)
• a sensor fails and gives wrong readings
• frequent readings of sensor values (e.g., each second/minute)
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Brewery Dataset
5
source: wikipedia
Fermenter
Temperature
sensors top
middle
bottom
Pressure
sensor
… and many more sensors…
and even more fermenters...
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Brewery Dataset
• From a company that produces more than two dozen craft beers
• Different beers can use different yeasts during fermentation and different fermenting temperatures
6
Stages of beer fermentation
From OSIsoft Users Conference 2017
OSIsoft: Introduction to Process
Optimization and Big-Data Analytics
with the PI System
https://www.youtube.com/watch?v=9Ad
vKkyTeBE
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Formally, a time series
7
• is an ordered sequence of values of a variable at equally spaced time intervals
• can be built on top of data obtained from sensors
– choose the size of time interval (e.g., 1 hour)
– represent all measurements in each hour by one value (e.g., average value / first value / last value …)
– we obtain a collection of values (e.g., one number per hour)
– the PI system can extract such values from collected data
• has many applications
– in forecasting, monitoring, or feedback and feed-forward control
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Multiple sources and “sampling” frequencies
8
Real life problem = multiple sensors
Manual = infrequent
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An illustrative Example
9
bottom
middle
top temperature sensor
Before doing analytics, we need to align the data streams (in this example, we could choose a multiple of one hour)!
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10
Task #1 Prediction of ADF
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The Problem Setting
11
• Automated measuring of gravity of wort is expensive
• The brewery measures it manually (and infrequently)
• It is used to compute Apparent Degree of Fermentation (ADF)
which is an important indicator of the status of fermentation
(Note: ADF indicates how much sugar has been converted to alcohol)
Our task: Predict ADF as well as possible eliminate a lot of manual work
improve quality of beer
decrease cost of fermentation
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Data and Challenges
12
• We can use the data from all sensors to predict ADF BUT
• We have only a few ADF values per fermentation cycle (and moreover it is not computed in “regular” intervals)
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The First Approach
13
• Let’s fix one beer brand, and plot our ADF points: x-axis - Time from start of fermentation y-axis - ADF
MyData <- read.csv(file="regression.txt")
MyData$time <- MyData$time/60/60
plot(MyData$time, MyData$adf, xlab =
"Time Since Fermentation [hours]",
ylab = "ADF", main = "Data without outliers")
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Simple Linear Regression Model
14
response (dependent)
variable
explanatory (independent)
variable
intercept slope
errors of the
response
variables should
be uncorrelated
with each other
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Linear Fit
15
Estimate
(Intercept) -0.0598497
Time 0.0159091
each hour, ADF
increases by
~0.016
plot(FilteredTimes, FilteredADF, xlab = "Time Since
Fermentation [hours]", ylab='ADF', main="ADF ~
Time")
linMod <- lm(FilteredADF ~ FilteredTimes)
abline(linMod$coefficients)
summary(linMod)
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Checking Assumptions with a Residual Plot
16
errors should be
independent of
predicted values
Our model is NOT ok!
predicted <- fitted(linMod)
errors <- resid(linMod)
plot(predicted, errors, main = "Residuals vs. predicted
ADF")
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What was wrong? How to improve it?
17
• We tried our luck with a simple linear model
• The model works OK for the data in the middle of our range, but is failing for small and large values of ADF
• As data scientists, we need to understand our problem domain.
• We need to learn about beer fermentation!
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Understanding the Fermentation Process (a bit better)
18
• What influences the speed of fermentation?
– temperature (we are cooling the wort, so this is relatively constant)
– amount of yeast
– amount of sugar and ethanol level
Gilliland, R.B., 1962. Yeast
reproduction during fermentation.
Journal of the Institute of Brewing,
68(3), pp.271-275.
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Speed of Fermentation
19
time http://wineserver.ucdavis.edu/industry/enology/fermentation_management/wine/problem_fermentations.html
sugar
content
One degree Brix
is 1 gram of
sucrose in 100
grams of solution
cells are adapting to the wort conditions and
engaging in cell division
fastest rate of fermentation, ethanol
level is relatively low
The rate slows because ethanol in
the medium forces an adaptation
of the plasma membrane. The
yeast can easily form a
membrane that depends upon
ethanol replacing water for its
structure and functionality…
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New (improved) model
20
• We now understand that there are different rates of fermentation.
• How about fitting 3 linear models? We would also need to figure out the break-points.
parameters to estimate
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intercept above zero
initial slope
Fitted Model
21
increase of slope: new
slope is 0.018011
decrease of slope: new
slope is 0.003764
break-points
xx <- FilteredTimes
yy <- FilteredADF
dati <- data.frame(x = xx, y = yy )
out.lm <- lm(y ~ x, data = dati)
o <- segmented(out.lm, seg.Z = ~x,
psi = list(x = c(10,40)),
control = seg.control(display = FALSE))
dat2 = data.frame(x = xx, y = broken.line(o)$fit)
o
Meaningful coefficients of the linear terms: (Intercept) x U1.x U2.x 0.005924 0.009560 0.008451 -0.014247 Estimated Break-Point(s): psi1.x psi2.x 14.57 45.61
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Fitted Model
22
ggplot(dati, aes(x = x, y = y)) + xlab("Time Since
Fermentation [hours]") +
ylab('ADF') +
geom_point() +
geom_line(data = dat2, color = 'blue')
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Checking Assumptions with a Residual Plot
23
errors <- resid(o)
plot( predicted, errors, main = "Residuals vs. predicted
ADF" )
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Task #2 Cooling prediction
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The Problem Setting
25
• After the fermentation (and possibly Free Rise or Diacetyl Rest) is finished, the beer has to be cooled almost to a freezing point
• The data contains temperature measurements at 3 different locations
Top
Middle
Bottom
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Typical Cooling Profile
26
In the ideal case, the
beer is cooling nicely
and all sensors read
similar values
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Atypical Cooling Profile
27
Top
Middle
Bottom
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All together
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Issues with data
29
• For each time point, we know in
which state of fermentation the
beer is (colors on graph)
• We also know about other sensors,
such as the cooling valves
• During fermentation, we try to
keep temperature constant
• The cooling is achieved by three
cooling valves (usually open 100%)
• However, there are many inconsistencies
Fermentation
Free rise
Diacetyl
Rest Cooling
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Issues with data
30
Cooling
labeled as
Diacetyl Rest
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Issues with data
31
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• We will ignore the “label” of the fermentation “stage”
• We define a cooling period simply as a uninterrupted interval with 100% cooling capacity
• Moreover, we will be interested in the data until the temperature reaches ~32F
Preprocessing
32
Free rise Cooling
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Filtered cooling profiles
33
● different initial
temperature for
various coolings
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Volumes
34
● various volumes of
beer in fermenter
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Thermodynamic heat transfer
35
• how the temperature will change in a small time interval ?
• real answer is hard and involves solving PDFs, …..
• we can try to approximate it
New temperature
Old temperature
Temperature
of “cooling”
substance
Cooling Rate ( <0 ) Volume
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Some simplifications
36
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Time difference = 30 min
37
Parameters:
Estimate Std. Error t value Pr(>|t|)
alpha -17.1918 0.3134 -54.86 <2e-16 ***
beta 30.4543 0.2427 125.49 <2e-16 ***
The cooling
substance is
~30.4F
d <- read.csv(file="UC2_TS_30.dat")
yOld <- d$o1
y <- d$n1
vol <- d$v
m <- nls(y ~ (1+a/vol)*yOld - a*b/vol)
summary(m)
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Residuals
38
we have more data when it is cooler
predicted <- fitted(m)
errors <- resid(m)
plot(predicted, errors) Residuals vs. Predicted Temperature
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Real cooling profile vs. Predicted cooling profile
39
real, measured data
computed from initial
temperature and volume
nd <- read.csv("UC2_Prediction.dat")
x<-seq(0,58*4,1)*0.25; y<-x*0
y[1] <- nd$m1[1]; vol <- nd$v[1]
alpha <- summary(m)$coefficients[1]
beta <- summary(m)$coefficients[2]
for (i in 2:length(x)) {
y[i] <- (1+alpha/vol)*y[i-1] - alpha*beta/vol
}
plot(nd$t/60, nd$m1, xlab = "Time Since Cooling [hours]",
ylab='Temperature [F]', type='l', col="red")
lines(x, y, type="s", col="blue")
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Task #3 Principal component analysis for
visualization
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Principal component analysis (PCA)
41
• PCA performs dimensionality reduction and de-correlation of dimensions
• The “real” data can lie within a smaller dimensional subspace
• PCA is useful for visualization and pre-processing
• Detect patterns and outliers
• Produce fewer and uncorrelated dimensions
• Good for data with relatively few labels
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Data preprocessing
42
• We have a lot of data from sensors; it is a bit tricky to use PCA directly (too many points, missing data)
• More importantly, we want to see when cycles might be same or differ
• We represented each fermentation cycle by 31 features, like
– highest ADF
– mean/min/max temperature during fermentation
– volume
– mean/min/max temperature during cooling
– duration of fermentation
– duration of cooling
– …...
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Visualization in 3D
43
how much variance is captured by the
ith principal component • We scale the data and compute the principal components
• We project the data onto two or three principal components that represent the most variance and plot them
# apply PCA - scale. = TRUE is highly
# advisable, but default is FALSE.
beers.pca <- prcomp(beers.data,
center = TRUE,
scale. = TRUE)
plot(beers.pca, type = "l")
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Visualization in 3D
44
• "Realtime Hops"
• "Red Wonder"
• "Alistair"
• "Kennedy"
• "Kerberos"
# Predict PCs
beers.projected <- predict(beers.pca,
newdata=beers.data)
scatterplot3d(beers.projected[,1],
beers.projected[,2],
beers.projected[,3],
color = beers.labels,
main = "PCA")
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Similar members
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Outliers
46
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Lessons Learned
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RESULTS CHALLENGE SOLUTION
COMPANY and GOAL
48
Lesson Learned
Predicting Apparent Degree of Fermentation (ADF)
Linear model of ADF as a function of time is insufficient
Use domain knowledge to develop a better model
Piecewise linear model is much improved over basic linear model
• Existence of three phases of fermentation suggests three segments in the linear model
• Errors are not correlated with predicted values
• Normality test values are better (not presented)
• Could consider other attributes
• ADF starts negative
• Errors are correlated with predicted values (BAD!)
• Normality test values are not near model (not presented)
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RESULTS CHALLENGE SOLUTION
COMPANY and GOAL
49
Lesson Learned
Predict cooling of fermented beer
Want to predict cooling trend, completion times
Basic thermodynamics function for cooling
Basic cooling model works well
• Use data to find values of constants
• Errors are not correlated with predicted values
• Could build complex model
• Many attributes available, including starting temperature, volume, and time
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RESULTS CHALLENGE SOLUTION
COMPANY and GOAL
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Lesson Learned
Visualizing high-dimensional data
Want to see similarities and differences between runs
Use PCA on run-focused data Can see outliers and patterns
• Represent situation of interest
• Identify reduced dimensionality
• Some beers are consistent
• Others include outliers
• Data in form of time series with many attributes
• Humans need at most 3-D data
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References for further study
• Gebhard Kirchgässner and Jürgen Wolters, Introduction to Modern Time Series Analysis, Springer Berlin Heidelberg New York, 2007.
• Peter J. Brockwell and Richard A. Davis, Introduction to Time Series and Forecasting, 2nd ed., Springer Texts in Statistics, Springer Science+Business Media, LLC, New York, NY, USA, 2002.
• James D. Hamilton, Time Series Analysis, Princeton University Press, Princeton, NJ, USA, 1994.
• NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, 2018.
• http://www.r-tutor.com
• http://wineserver.ucdavis.edu/industry/enology/fermentation_management/wine/problem_fermentations.html
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Contact Information
Brian D. Davison [email protected]
Associate Prof., Lehigh University
Slides and R source code available from:
http://www.cse.lehigh.edu/~brian/PIW18/ Lehigh University's Linderman Library
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